Characterization of complex interdependencies of oscillatory processes from data with extended phase dynamics methods

• Applicant: Professor Dr. Arkady Pikovsky
• Subject Area: General, Cognitive and Mathematical Psychology
• Term: from 2007 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 34181293

• Final Report Year: 2016

Final Report Abstract

In the research work within the framework of the Project B3 we made an essential progress in developing the coupled-oscillator approach to multivariate data analysis, introduced in our previous publications. This analysis deals with a network of coupled self-sustained oscillators. The coupling shall be not too strong, so that the oscillators remain asynchronous. This limitation is relaxed in typical case of noisy systems, because noisy perturbations drive the systems away from the synchronous state. The analysis provides information on the directional connectivity by means of reconstruction of phase dynamics equation and analysis of coupling functions. Therefore the results reflect the effective phase connectivity, which is close bout not identical with the structural connectivity. The additional (structurally non-existing links) are not spurious but appear due to nonlinear non-pairwise coupling or due to high-order terms in the phase dynamics approximation. These links increasingly appear with increase of the coupling strength. We pursued the development of the approach in two main directions. First, we elaborated on the twodimensional reconstruction. Here we have exploited the kernel density estimation. This statistical approach provides reliable coupling function estimation in case of relatively short data or when the interacting systems are close to synchrony. Next, we introduced a technique for decomposing the coupling function into a product of PRC and forcing function. In this way we made possible determination of PRC, which is the most important characteristic of an oscillator, without isolation of the system from its environment and without application of specially designed stimulation. We illustrated this novel technique by computation of PRC for the human cardiac oscillator, coupled to the respiratory system. The second direction was related to extension of the technique to the case of more than two oscillators. Typically, in this case one performs only pair-wise analysis. We have shown, that the connectivity reconstruction can be essentially enhanced by analysis of triplets of oscillators. For this purpose we quantify strength of a connection by taking into account not only two oscillators, connected with this link, but also all other elements of the network. In this way we correctly describe effects due to non-direct connections, common drive, etc. We stress that although the method is based on the phase dynamics equations, it is not restricted to the case of very weak coupling. Indeed, analytical derivation of these equations requires weakness of coupling, but numerical reconstruction is possible as long as the signals can be considered as nearly periodic or weakly chaotic. Certainly, application of the method implies that the outputs of all nodes, suitable for phase estimation, are available. An important issue is that phase dynamics equations are valid also for transient processes. Hence, the techniques does not imply stationarity of the data. So, e.g., if the network is repeatedly stimulated, the pieces of data between the stimuli can be used for the phase dynamics reconstruction and, therefore, characterization of the network connectivity. Moreover, repeated perturbation can be a useful tool in case when the network (or some nodes) are close to synchrony, so that the reconstruction without perturbation fails. Perturbing the system and observing its relaxation to the synchronous state one can obtain enough data for successful application of the technique. This feature makes the technique suitable for event-related analysis. Indeed, if a single evoked response is too short, one can use for averaging data obtained in several trials.

Publications

• (2013). In vivo cardiac phase response curve elucidates human respiratory heart rate variability. Nature Communications,, 4:2418
  Kralemann, B., Frühwirth, M., Pikovsky, A., Rosenblum, M., Kenner, T., Schaefer, J., and Moser, M.
  
• (2014). How much time has passed? Ask your heart. Frontiers in Neurorobotics, 8:15
  Pollatos, O., Yeldesbay, A., Pikovsky, A., and Rosenblum, M.
  
• (2014). Reconstructing effective phase connectivity of oscillator networks from observations. New Journal of Physics, 16:085013
  Kralemann, B., Pikovsky, A., and Rosenblum, M.